Lectures on Arithmetic Noncommutative Geometrymarcolli/BookVanderbilt.pdf · 2004. 9. 28. ·...

Lectures on Arithmetic

Noncommutative Geometry

Matilde Marcolli

And indeed there will be timeTo wonder “Do I dare?” and, “Do I dare?”Time to turn back and descend the stair.

...

Do I dareDisturb the Universe?

...

For I have known them all already, known them all;Have known the evenings, mornings, afternoons,I have measured out my life with coffee spoons.

...

I should have been a pair of ragged clawsScuttling across the floors of silent seas....No! I am not Prince Hamlet, nor was meant to be;Am an attendant lord, one that will doTo swell a progress, start a scene or two...At times, indeed, almost ridiculous–Almost, at times, the Fool....We have lingered in the chambers of the seaBy sea-girls wreathed with seaweed red and brownTill human voices wake us, and we drown.

(T.S. Eliot, “The Love Song of J. Alfred Prufrock”)

Contents

Chapter 1. Ouverture 51. The NCG dictionary 72. Noncommutative spaces 83. Spectral triples 9

Chapter 2. Noncommutative modular curves 171. Modular curves 172. The noncommutative boundary of modular curves 243. Modular interpretation: noncommutative elliptic curves 244. Limiting modular symbols 295. Hecke eigenforms 406. Selberg zeta function 437. The modular complex and K-theory of C∗-algebras 448. Intermezzo: Chaotic Cosmology 45

Chapter 3. Quantum statistical mechanics and Galois theory 531. Quantum Statistical Mechanics 542. The Bost–Connes system 583. Noncommutative Geometry and Hilbert’s 12th problem 634. The GL2 system 655. Quadratic fields 72

Chapter 4. Noncommutative geometry at arithmetic infinity 811. Schottky uniformization 812. Dynamics and noncommutative geometry 883. Arithmetic infinity: archimedean primes 934. Arakelov geometry and hyperbolic geometry 965. Intermezzo: Quantum gravity and black holes 1006. Dual graph and noncommutative geometry 1047. Arithmetic varieties and L–factors 1098. Archimedean cohomology 115

Bibliography 125

3

CHAPTER 1

Ouverture

Noncommutative geometry, as developed by Connes starting in theearly ’80s ([15], [17], [20]), extends the tools of ordinary geometry totreat spaces that are quotients, for which the usual “ring of functions”,defined as functions invariant with respect to the equivalence relation,is too small to capture the information on the “inner structure” ofpoints in the quotient space. Typically, for such spaces functions on thequotients are just constants, while a nontrivial ring of functions, whichremembers the structure of the equivalence relation, can be definedusing a noncommutative algebra of coordinates, analogous to the non-commuting variables of quantum mechanics. These “quantum spaces”are defined by extending the Gelfan’d–Naimark correspondence

X loc.comp. Hausdorff space⇔ C0(X) abelian C∗-algebraby dropping the commutativity hypothesis in the right hand side. Thecorrespondence then becomes a definition of what is on the left handside: a noncommutative space.

Such quotients are abundant in nature. They arise, for instance,from foliations. Several recent results also show that noncommuta-tive spaces arise naturally in number theory and arithmetic geometry.The first instance of such connections between noncommutative geom-etry and number theory emerged in the work of Bost and Connes [8],which exhibits a very interesting noncommutative space with remark-able arithmetic properties related to class field theory. This reveals avery useful dictionary that relates the phenomena of spontaneous sym-metry breaking in quantum statistical mechanics to the mathematics ofGalois theory. This space can be viewed as the space of 1-dimensionalQ-lattices up to scale, modulo the equivalence relation of commensu-rability (cf. [29]). This space is closely related to the noncommutativespace used by by Connes to obtain a spectral realization of the zerosof the Riemann zeta function, [22]. In fact, this is again the spaceof commensurability classes of 1-dimensional Q-lattices, but with thescale factor also taken into account.

More recently, other results that point to deep connections betweennoncommutative geometry and number theory appeared in the work

5

6 1. OUVERTURE

of Connes and Moscovici [33] [34] on the modular Hecke algebras.This shows that the Rankin–Cohen brackets, an important algebraicstructure on modular forms [94], have a natural interpretation in thelanguage of noncommutative geometry, in terms of the Hopf algebra ofthe transverse geometry of codimension one foliations. The modularHecke algebras, which naturally combine products and action of Hecleoperators on modular forms, can be viewed as the “holomorphic part”of the algebra of coordinates on the space of commensurability classesof 2-dimensional Q-lattices constructed in joint work of Connes andthe author [29].

Cases of occurrences of interesting number theory within noncom-mutative geometry can be found in the classification of noncommuta-tive three spheres by Connes and Dubois–Violette [26] [27]. Here thecorresponding moduli space has a ramified cover by a noncommutativenilmanifold, where the noncommutative analog of the Jacobian of thiscovering map is expressed naturally in terms of the ninth power of theDedekind eta function. Another such case occurs in Connes’ calcula-tion [24] of the explicit cyclic cohomology Chern character of a spectraltriple on SUq(2) defined by Chakraborty and Pal [12].

Other instances of noncommutative spaces that arise in the contextof number theory and arithmetic geometry can be found in the non-commutative compactification of modular curves of [25], [70]. Thisnoncommutative space is again related to the noncommutative geom-etry of Q-lattices. In fact, it can be seen as a stratum in the com-pactification of the space of commensurability classes of 2-dimensionalQ-lattices (cf. [29]).

Another context in which noncommutative geometry provides a use-ful tool for arithmetic geometry is in the description of the totallydegenerate fibers at “arithmetic infinity” of arithmetic varieties overnumber fields, analyzed in joint work of the author with Katia Consani([36], [37], [38], [39]).

The present text is based on a series of lectures given by the authorat Vanderbilt University in May 2004, as well as on previous series oflectures given at the Fields Institute in Toronto (2002), at the Univer-sity of Nottingham (2003), and at CIRM in Luminy (2004).

The main focus of the lectures is the noncommutative geometryof modular curves (following [70]) and of the archimedean fibers ofarithmetic varieties (following [36]). A chapter on the noncommutativespace of commensurability classes of 2-dimensional Q-lattices is alsoincluded (following [29]). The text reflects very closely the style ofthe lectures. In particular, we have tried more to convey the generalpicture than the details of the proofs of the specific results. Though

1. THE NCG DICTIONARY 7

many proofs have not been included in the text, the reader will findreferences to the relevant literature, where complete proofs are provided(in particular [29], [36], and [70]).

More explicitly, the text is organized as follows:

• We start by recalling a few preliminary notions of noncommu-tative geometry (following [20]).• The second chapter describes how various arithmetic proper-

ties of modular curves can be seen by their “noncommutativeboundary”. This part is based on the joint work of Yuri Maninand the author. The main references are [70], [71], [72].• We review briefly the work of Connes and the author [29] on

the noncommutative geometry of commensurability classes ofQ-lattices. The relation of the noncommutative space of com-mensurability classes of Q-lattices to the Hilbert 12th prob-lem of explicit class field theory is based on ongoing work ofConnes, Ramachandran and the author [31], on the originalwork of Bost and Connes [8] and on Manin’s real multiplica-tion program [62] [63].• The noncommutative geometry of the fibers at “arithmetic in-

finity” of varieties over number fields is based on joint work ofConsani and the author, for which the references are [36], [37],[38], [39], [40]. This chapter also contains a detailed accountof Manin’s formula for the Green function of Arakelov geom-etry for arithmetic surfaces, based on [66], and a proposedphysical interpretation of this formula, as in [69].

1. The NCG dictionary

There is a dictionary (cf. [20]) relating concepts of ordinary geom-etry to the corresponding counterparts in noncommutative geometry.The entries can be arranged according to the finer structures consideredon the underlying space, roughly according to the following table.

measure theory von Neumann algebrastopology C∗–algebras

smooth structures smooth subalgebrasRiemannian geometry spectral triples

It is important to notice that, usually, the notions of noncommuta-tive geometry are “richer” than the corresponding entries of the dictio-nary on the commutative side. For instance, as Connes discovered, non-commutative measure spaces (von Neumann algebras) come endowedwith a natural time evolution which is trivial in the commutative case.

8 1. OUVERTURE

Similarly, at the level of topology one often sees phenomena that arecloser to rigid analytic geometry. This is the case, for instance, with thenoncommutative tori Tθ, which already at the C

∗-algebra level exhibitmoduli that behave much like moduli of one-dimensional complex tori(elliptic curves) in the commutative case.

In the context we are going to discuss this richer structure of non-commutative spaces is crucial, as it permits us to use tools like C∗-algebras (topology) to study the properties more rigid spaces like alge-braic or arithmetic varieties.

2. Noncommutative spaces

The way to assign the algebra of coordinates to a quotient spaceX = Y/ ∼ can be explained in a short slogan as follows:

• Functions on Y with f(a) = f(b) for a ∼ b. Poor!• Functions fab on the graph of the equivalence relation. Good!

The second description leads to a noncommutative algebra, as theproduct, determined by the groupoid law of the equivalence relation,has the form of a convolution product (like the product of matrices).

For sufficiently nice quotients, even though the two notions arenot the same, they are related by Morita equivalence, which is thesuitable notion of “isomorphism” between noncommutative spaces. Formore general quotients, however, the two notions truly differ and thesecond one is the only one that allows one to continue to make senseof geometry on the quotient space.

A very simple example illustrating the above situation is the fol-lowing (cf. [23]). Consider the topological space Y = [0, 1] × {0, 1}with the equivalence relation (x, 0) ∼ (x, 1) for x ∈ (0, 1). By the firstmethod one only obtains constant functions C, while by the secondmethod one obtains

{f ∈ C([0, 1])⊗M2(C) : f(0) and f(1) diagonal }which is an interesting nontrivial algebra.

The idea of preserving the information on the structure of the equiv-alence relation in the description of quotient spaces has analogs inGrothendieck’s theory of stacks in algebraic geometry.

2.1. Morita equivalence. In noncommutative geometry, isomor-phisms of C∗-algebras are too restrictive to provide a good notion ofisomorphisms of noncommutative spaces. The correct notion is pro-vided by Morita equivalence of C∗-algebras.

We have equivalent C∗-algebras A1 ∼ A2 if there exists a bimoduleM, which is a right HilbertA1 module with anA1-valued inner product

3. SPECTRAL TRIPLES 9

〈·, ·〉A1, and a left Hilbert A2-module with an A2-valued inner product〈·, ·〉A2, such that we have:

• We obtain all Ai as the closure of the span of{〈ξ1, ξ2〉Ai : ξ1, ξ2 ∈ M}.

• ∀ξ1, ξ2, ξ3 ∈ M we have〈ξ1, ξ2〉A1ξ3 = ξ1〈ξ2, ξ3〉A2 .

• A1 and A2 act onM by bounded operators,〈a2ξ, a2ξ〉A1 ≤ ‖a2‖2〈ξ, ξ〉A1 〈a1ξ, a1ξ〉A2 ≤ ‖a1‖2〈ξ, ξ〉A2

for all a1 ∈ A1, a2 ∈ A2, ξ ∈ M.This notion of equivalence roughly means that one can transfer

modules back and forth between the two algebras.

2.2. The tools of noncommutative geometry. Once one iden-tifies in a specific problem a space that, by its nature of quotient of thetype described above, is best described as a noncommutative space,there is a large set of well developed techniques that one can use tocompute invariants and extract essential information from the geome-try. The following is a list of some such techniques, some of which willmake their appearance in the cases treated in these notes.

• Topological invariants: K-theory• Hochschild and cyclic cohomology• Homotopy quotients, assembly map (Baum-Connes)• Metric structure: Dirac operator, spectral triples• Characteristic classes, zeta functions

We will recall the necessary notions when needed. We now beginby taking a closer look at the analog in the noncommutative world ofRiemannian geometry, which is provided by Connes’ notion of spectraltriples.

3. Spectral triples

Spectral triples are a crucial notion in noncommutative geometry.They provide a powerful and flexible generalization of the classicalstructure of a Riemannian manifold. The two notions agree on a com-mutative space. In the usual context of Riemannian geometry, thedefinition of the infinitesimal element ds on a smooth spin manifoldcan be expressed in terms of the inverse of the classical Dirac opera-tor D. This is the key remark that motivates the theory of spectraltriples. In particular, the geodesic distance between two points on themanifold is defined in terms of D−1 (cf. [20] §VI). The spectral triple

10 1. OUVERTURE

(A, H, D) that describes a classical Riemannian spin manifold is givenby the algebra A of complex valued smooth functions on the manifold,the Hilbert space H of square integrable spinor sections, and the clas-sical Dirac operator D. These data determine completely and uniquelythe Riemannian geometry on the manifold. It turns out that, whenexpressed in this form, the notion of spectral triple extends to moregeneral non-commutative spaces, where the data (A, H, D) consist ofa C∗-algebra A (or more generally of some smooth subalgebra of a C∗-algebra) with a representation in the algebra of bounded operators ona separable Hilbert space H, and an operator D on H that verifies themain properties of a Dirac operator.

We recall the basic setting of Connes theory of spectral triples. Fora more complete treatment see [20], [21], [32].

Definition 3.1. A spectral triple (A,H, D) consists of a C∗-algebraA with a representation

ρ : A → B(H)in the algebra of bounded operators on a separable Hilbert space H, andan operator D (called the Dirac operator) on H, which satisfies thefollowing properties:

(1) D is self–adjoint.(2) For all λ /∈ R, the resolvent (D − λ)−1 is a compact operator

on H.(3) The commutator [D, ρ(a)] is a bounded operator on H, for all

a ∈ A0 ⊂ A, a dense involutive subalgebra of A.The property 2. of Definition 3.1 can be regarded as a general-

ization of the ellipticity property of the standard Dirac operator on acompact manifold. In the case of ordinary manifolds, we can consideras subalgebra A0 the algebra of smooth functions, as a subalgebra ofthe commutative C∗-algebra of continuous functions. In fact, in theclassical case of Riemannian manifolds, property 3. is equivalent theLipschitz condition, hence it is satisfied by a larger class than that ofsmooth functions.

Thus, the basic geometric structure encoded by the theory of spec-tral triples is Riemannian geometry, but in more refined cases, suchas Kähler geometry, the additional structure can be easily encoded asadditional symmetries. We will see, for instance, a case (cf. [36] [40])where the algebra involves the action of the Lefschetz operator of acompact Kähler manifold, hence it encodes the information (at thecohomological level) on the Kähler form.


Since we are mostly interested in the relations between noncom-mutative geometry and arithmetic geometry and number theory, anespecially interesting feature of spectral triples is that they have anassociated family of zeta functions and a theory of volumes and inte-gration, which is related to special values of these zeta functions. (Thefollowing treatment is based on [20], [21].)

3.1. Volume form. A spectral triple (A,H, D) is said to be ofdimension n, or n–summable if the operator |D|−n is an infinitesimalof order one, which means that the eigenvalues λk(|D|−n) satisfy theestimate λk(|D|−n) = O(k−1).

For a positive compact operator T such that

k−1∑

j=0

λj(T ) = O(log k),

the Dixmier trace Trω(T ) is the coefficient of this logarithmic diver-gence, namely

(1.1) Trω(T ) = limω

1

log k

k∑

j=1

λj(T ).

Here the notation limω takes into account the fact that the sequence

S(k, T ) :=1

log k

k∑

j=1

λj(T )

is bounded though possibly non-convergent. For this reason, the usualnotion of limit is replaced by a choice of a linear form limω on theset of bounded sequences satisfying suitable conditions that extendanalogous properties of the limit. When the sequence S(k, T ) converges(1.1) is just the ordinary limit Trω(T ) = limk→∞ S(k, T ). So defined,the Dixmier trace (1.1) extends to any compact operator that is aninfinitesimal of order one, since any such operator is the difference oftwo positive ones. The operators for which the Dixmier trace doesnot depend on the choice of the linear form limω are called measurableoperators.

On a non-commutative space the operator |D|−n generalizes thenotion of a volume form. The volume is defined as

(1.2) V = Trω(|D|−n).More generally, consider the algebra Ã generated by A and [D,A].Then, for a ∈ Ã, integration with respect to the volume form |D|−n is

12 1. OUVERTURE

defined as

(1.3)

∫

a :=1

VTrω(a|D|−n).

The usual notion of integration on a Riemannian spin manifold Mcan be recovered in this context (cf. [20], [58]) through the formula (neven):

∫

M

fdv =(

2n−[n/2]−1πn/2nΓ(n/2))

Trω(f |D|−n).Here D is the classical Dirac operator on M associated to the metricthat determines the volume form dv, and f in the right hand side isregarded as the multiplication operator acting on the Hilbert space ofsquare integrable spinors on M .

3.2. Zeta functions. An important function associated to theDirac operator D of a spectral triple (A,H, D) is its zeta function(1.4) ζD(z) := Tr(|D|−z) =

∑

λ

Tr(Π(λ, |D|))λ−z,

where Π(λ, |D|) denotes the orthogonal projection on the eigenspaceE(λ, |D|).

An important result in the theory of spectral triples ([20] §IVProposition 4) relates the volume (1.2) with the residue of the zetafunction (1.4) at s = 1 through the formula

(1.5) V = lims→1+

(s− 1)ζD(s) = Ress=1Tr(|D|−s).

There is a family of zeta functions associated to a spectral triple(A,H, D), to which (1.4) belongs. For an operator a ∈ Ã, we candefine the zeta functions

(1.6) ζa,D(z) := Tr(a|D|−z) =∑

λ

Tr(a Π(λ, |D|))λ−z

and

(1.7) ζa,D(s, z) :=∑

λ

Tr(a Π(λ, |D|))(s− λ)−z.

These zeta functions are related to the heat kernel e−t|D| by Mellintransform

(1.8) ζa,D(z) =1

Γ(z)

∫ ∞

0

tz−1Tr(a e−t|D|) dt

where

(1.9) Tr(a e−t|D|) =∑

λ

Tr(a Π(λ, |D|))e−tλ =: θa,D(t).


Similarly,

(1.10) ζa,D(s, z) =1

Γ(z)

∫ ∞

0

θa,D,s(t) tz−1 dt

with

(1.11) θa,D,s(t) :=∑

λ

Tr(a Π(λ, |D|))e(s−λ)t.

Under suitable hypothesis on the asymptotic expansion of (1.11) (cf.Theorem 2.7-2.8 of [65] §2), the functions (1.6) and (1.7) admit aunique analytic continuation (cf. [32]) and there is an associated reg-ularized determinant in the sense of Ray–Singer (cf. [84]):

(1.12) det∞ a,D

(s) := exp

(

− ddz

ζa,D(s, z)|z=0)

.

The family of zeta functions (1.6) also provides a refined notion ofdimension for a spectral triple (A,H, D), called the dimension spec-trum. This is a subset Σ = Σ(A,H, D) in C with the property thatall the zeta functions (1.6), as a varies in Ã, extend holomorphicallyto C \ Σ.

Examples of spectral triples with dimension spectrum not containedin the real line can be constructed out of Cantor sets.

3.3. Index map. The data of a spectral triple determine an indexmap. In fact, the self adjoint operator D has a polar decomposition,D = F |D|, where |D| =

√D2 is a positive operator and F is a sign

operator, i.e. F 2 = I. Following [20] and [21] one defines a cycliccocycle

(1.13) τ(a0, a1, . . . , an) = Tr(a0[F, a1] · · · [F, an]),where n is the dimension of the spectral triple. For n even Tr shouldbe replaced by a super trace, as usual in index theory. This cocyclepairs with the K-groups of the algebra A (with K0 in the even case andwith K1 in the odd case), and defines a class τ in the cyclic cohomologyHCn(A) of A. The class τ is called the Chern character of the spectraltriple (A,H, D).

In the case when the spectral triple (A,H, D) has discrete dimen-sion spectrum Σ, there is a local formula for the cyclic cohomologyChern character (1.13), analogous to the local formula for the indexin the commutative case. This is obtained by producing Hochschildrepresentatives (cf. [21], [32])

(1.14) ϕ(a0, a1, . . . , an) = Trω(a0[D, a1] · · · [D, an] |D|−n).

14 1. OUVERTURE

3.4. Infinite dimensional geometries. The main difficulty inconstructing specific examples of spectral triples is to produce an op-erator D that at the same time has bounded commutators with theelements of A and produces a non-trivial index map.

It sometimes happens that a noncommutative space does not admita finitely summable spectral triple, namely one such that the operator|D|−p is of trace class for some p > 0. Obstructions to the existence ofsuch spectral triples are analyzed in [19]. It is then useful to considera weaker notion, namely that of θ-summable spectral triples. Thesehave the property that the Dirac operator satisfies

(1.15) Tr(e−tD2

) 0.Such spectral triples should be thought of as “infinite dimensional non-commutative geometries”.

We’ll see examples of spectral triples that are θ-summable but notfinitely summable, because of the growth rate of the multiplicities ofeigenvalues.

3.5. Spectral triples and Morita equivalences. If (A1,H, D)is a spectral triple, and we have a Morita equivalence A1 ∼ A2 im-plemented by a bimodule M which is a finite projective right Hilbertmodule over A1, then we can transfer the spectral triple from A1 toA2.

First consider the A1-bimodule Ω1D generated by{a1[D, b1] : a1, b1 ∈ A1}.

We define a connection

∇ :M→M⊗A1 Ω1Dby requiring that

∇(ξa1) = (∇ξ)a1 + ξ ⊗ [D, a1],∀ξ ∈ M, ∀a1 ∈ A1 and ∀ξ1, ξ2 ∈ M. We also require that

〈ξ1,∇ξ2〉A1 − 〈∇ξ1, ξ2〉A1 = [D, 〈ξ1, ξ2〉A1 ].This induces a spectral triple (A2, H̃, D̃) obtained as follows.The Hilbert space is given by H̃ =M⊗A1 H. The action takes the

form

a2 (ξ ⊗A1 x) := (a2ξ)⊗A1 x.The Dirac operator is given by

D̃(ξ ⊗ x) = ξ ⊗D(x) + (∇ξ)x.


Notice that we need a Hermitian connection ∇, because commuta-tors [D, a] for a ∈ A1 are non-trivial, hence 1 ⊗ D would not be welldefined on H̃.

3.6. K-theory of C∗-algebras. The K-groups are important in-variants of C∗-algebras that capture information on the topology ofnon-commutative spaces, much like cohomology (or more appropriatelytopological K-theory) captures information on the topology of ordinaryspaces. For a C∗-algebra A, we have:

3.6.1. K0(A). This group is obtained by considering idempotents(p2 = p) in matrix algebras over A. Notice that, for a C∗-algebra, itis sufficient to consider projections (P 2 = P , P = P ∗). In fact, wecan always replace an idempotent p by a projection P = pp∗(1− (p−p∗)2)−1, preserving the von Neumann equivalence. Then we considerP ' Q if and only if P = X∗X Q = XX∗, for X = partial isometry(X = XX∗X), and we impose the stable equivalence: P ∼ Q, forP ∈ Mn(A) and Q ∈ Mm(A), if and only if there exists R a projectionwith P⊕R ' Q⊕R. We define K0(A)+ to be the monoid of projectionsmodulo these equivalences and we let K0(A) be its Grothendieck group.

3.6.2. K1(A). Consider the group GLn(A) of invertible elementsin the matrix algebraMn(A), and GL0n(A) the identity component ofGLn(A). The morphism

GLn(A)→ GLn+1(A) a 7→(

a 00 1

)

inducesGLn(A)/GL0n(A)→ GLn+1(A)/GL0n+1(A).

The group K1(A) is defined as the direct limit of these morphisms.Notice that K1(A) is abelian even if the GLn(A)/GL0n(A) are not.

In general, K-theory is not easy to compute. This lead to twofundamental developments in noncommutative geometry. The first iscyclic cohomology (cf. [17], [18], [20]), introduced by Connes in 1981,which provides cycles that pair with K-theory, and a Chern charac-ter. The second development is the geometrically defined K-theory ofBaum–Connes [6] and the assembly map

(1.16) µ : K∗(X, G)→ K∗(C0(X) o G)from the geometric to the analytic K-theory. Much work went, inrecent years, into exploring the range of validity of the Baum–Connesconjecture, according to which the assembly map gives an isomorphismbetween these two different notions of K-theory.

CHAPTER 2

Noncommutative modular curves

The results of this section are mostly based on the joint work ofYuri Manin and the author [70], with additional material from [71]and [72]. We include necessary preliminary notions on modular curvesand on noncommutative tori, respectively based on the papers [68] and[15], [85].

We first recall some aspects of the classical theory of modularcurves. We will then show that these notions can be entirely recoveredand enriched through the analysis of a noncommutative space associ-ated to a compactification of modular curves, which includes noncom-mutative tori as possible degenerations of elliptic curves.

1. Modular curves

Let G be a finite index subgroup of the modular group Γ = PSL(2, Z),and let XG denote the quotient

(2.1) XG := G\H2

where H2 is the 2-dimensional real hyperbolic plane, namely the upperhalf plane {z ∈ C : =z > 0} with the metric ds2 = |dz|2/(=z)2.Equivalently, we identify H2 with the Poincaré disk {z : |z| < 1} withthe metric ds2 = 4|dz|2/(1− |z|2)2.

We denote the quotient map by φ : H2 → XG.Let P denote the coset space P := Γ/G. We can write the quotient

XG equivalently as

XG = Γ\(H2 × P).A tessellation of the hyperbolic plane H2 by fundamental domains

for the action of PSL(2, Z) is illustrated in Figure 1. The action ofPSL(2, Z) by fractional linear transformations z 7→ az+b

cz+dis usually

written in terms of the generators S : z 7→ −1/z and T : z 7→ z + 1(inversion and translation). Equivalently, PSL(2, Z) can be identifiedwith the free product PSL(2, Z) ∼= Z/2 ∗ Z/3, with generators σ, τ oforder two and three, respectively given by σ = S and τ = ST .

17

18 2. NONCOMMUTATIVE MODULAR CURVES

Figure 1. Fundamental domains for PSL(2, Z)

An example of finite index subgroups is given by the congruencesubgroups Γ0(N) ⊂ Γ, of matrices

(

a bc d

)

with c ≡ 0 mod N . A fundamental domain for Γ0(N) is given byF ∪N−1k=0 ST k(F ), where F is a fundamental domain for Γ as in Figure11.

The quotient space XG has the structure of a non-compact Riemannsurface. This has a natural algebro-geometric compactification, whichconsists of adding the cusp points (points at infinity). The cusp pointsare identified with the quotient

(2.2) G\P1(Q) ' Γ\(P1(Q)× P).Thus, we write the compactification as

(2.3) XG := G\(H2 ∪ P1(Q)) ' Γ\(

(H2 ∪ P1(Q))× P)

.

The modular curve XΓ, for Γ = SL(2, Z) is the moduli space ofelliptic curves, with the point τ ∈ H2 parameterizing the lattice Λ =Z⊕ τZ in C and the corresponding elliptic curve uniformized by

Eτ = C/Λ.

1Figures 1 and 2 are taken from Curt McMullen’s Gallery

1. MODULAR CURVES 19

The unique cusp point corresponds to the degeneration of the ellipticcurve to the cylinder C∗, when τ →∞ in the upper half plane.

The other modular curves, obtained as quotients XG by a con-gruence subgroup, can also be interpreted as moduli spaces: they aremoduli spaces of elliptic curves with level structure. Namely, for ellip-tic curves E = C/Λ, this is an additional information on the torsionpoints 1

NΛ/Λ ⊂ QΛ/Λ of some level N .

For instance, in the case of the principal congruence subgroupsΓ(N) of matrices

M =

(

a bc d

)

∈ Γ

such that M ≡ Id mod N , points in the modular curve Γ(N)\H2classify elliptic curves Etau = C/Λ, with Λ = Z + Zτ , together witha basis {1/N, τ/N} for the torsion subgroup 1

NΛ/Λ. The projection

XΓ(N) → XΓ forgets the extra structure.In the case of the groups Γ0(N), points in the quotient Γ0(N)\H2

classify elliptic curves together with a cyclic subgroup of Eτ of orderN . This extra information is equivalent to an isogeny φ : Eτ → Eτ ′where the cyclic group is Ker(φ). Recall that an isogeny is a morphismφ : Eτ → Eτ ′ such that φ(0) = 0. These are implemented by the actionof GL+2 (Q) on H

2, namely Eτ and Eτ ′ are isogenous if and only if τand τ ′ in H2 are in the same orbit of GL+2 (Q).

1.1. Modular symbols. Given two points α, β ∈ H2 ∪ P1(Q), areal homology class {α, β}G ∈ H1(XG, R) is defined as

{α, β}G : ω 7→∫ β

α

φ∗(ω),

where ω are holomorphic differentials on XG, and the integration of thepullback to H2 is along the geodesic arc connecting two cusps α and β(cf. Figure 2).

The modular symbols {α, β}G satisfy the additivity and invarianceproperties

{α, β}G + {β, γ}G = {α, γ}G,and

{gα, gβ}G = {α, β}G,for all g ∈ G.

Because of additivity, it is sufficient to consider modular symbolsof the form {0, α} with α ∈ Q,

{0, α}G = −n∑

k=1

{gk(α) · 0, gk(α) · i∞}G,


Figure 2. Geodesics between cusps define modularsymbols

where α has continued fraction expansion α = [a0, . . . , an], and

gk(α) =

(

pk−1(α) pk(α)qk−1(α) qk(α)

)

,

with pk/qk the successive approximations, and pn/qn = α.

In the classical theory of the modular symbols of [68], [77] (cf. also[50], [76]) cohomology classes obtained from cusp forms are evaluatedagainst relative homology classes given by modular symbols, namely,given a cusp form Φ on H, obtained as pullback Φ = ϕ∗(ω)/dz underthe quotient map ϕ : H → XG, we denote by ∆ω(s) the intersectionnumbers ∆ω(s) =

∫ gs(i∞)

gs(0)Φ(z)dz, with gsG = s ∈ P. These intersection

numbers can be interpreted in terms of special values of L–functionsassociated to the automorphic form which determines the cohomologyclass.

We rephrase this in cohomological terms following [77]. We denoteby Ĩ and R̃ the elliptic points, namely the orbits Ĩ = Γ · i and R̃ = Γ ·ρ.We denote by I and R the image in XG of the elliptic points

I = G\Ĩ R = G\R̃,with ρ = eπi/3. We use the notation

(2.4) HBA := H1(XG \ A, B; Z).


These groups are related by the pairing

(2.5) HBA ×HAB → Z.The modular symbols {g(0), g(i∞)}, for gG ∈ P, define classes in

Hcusps. For σ and τ the generators of PSL(2, Z) with σ2 = 1 and τ 3 = 1,we set

(2.6) PI = 〈σ〉\P and PR = 〈τ〉\P.There is an isomorphism Z|P| ∼= HRcusps∪I . Given the exact sequences

0→ Hcusps ι′

→ HRcuspsπR→ Z|PR| → Z→ 0

and0→ Z|PI | → HRcusps∪I

πI→ HRcusps → 0,the image πI(x̃) ∈ HRcusps of an element x̃ =

∑

s∈P λss in Z|P| ∼= HRcusps∪I

represents an element x ∈ Hcusps iff the image πR(πI(x̃)) = 0 in Z|PR|.As proved in [77], for s = gG ∈ P, the intersection pairing • : H cusps ×Hcusps → Z gives

{g(0), g(i∞)} • x = λs − λσs.Thus, we write the intersection number as a function ∆x : P→ R by(2.7) ∆x(s) = λs − λσs,where x is given as above.

1.2. The modular complex. For x and y in H2 we denote by〈x, y〉 the oriented geodesic arc connecting them. Moreover, in thedecomposition PSL(2, Z) = Z/2 ∗ Z/3 as a free product, we denote byσ the generator of Z/2 and by τ the generator of Z/3.

Definition 1.1. The modular complex is the cell complex definedas follows.

• 0-cells: the cusps G\P1(Q), and the elliptic points I and R.• 1-cells: the oriented geodesic arcs G\(Γ · 〈i∞, i〉) and G\(Γ ·〈i, ρ〉), where by Γ· we mean the orbit under the action of Γ.• 2-cells: G\{Γ·E}, where E is the polygon with vertices {i, ρ, 1+

i, i∞} and sides the corresponding geodesic arcs.• Boundary operator: ∂ : C2 → C1 is given by

gE 7→ g〈i, ρ〉+ g〈ρ, 1 + i〉 + g〈1 + i, i∞〉+ g〈i∞, i〉,for g ∈ Γ, and the boundary ∂ : C1 → C0 is given by

g〈i∞, i〉 7→ g(i)− g(i∞)g〈i, ρ〉 7→ g(ρ)− g(i).


Figure 3. The cell decomposition of the modular com-plex

This gives a cell decomposition of H2 adapted to the action ofPSL(2, Z) and congruence subgroups, cf. Figure 3.

We have the following result [68].

Proposition 1.1. The modular complex computes the first homol-ogy of XG:

(2.8) H1(XG) ∼=Ker(∂ : C1 → C0)Im(∂ : C2 → C1)

.

We can derive versions of the modular complex that compute rela-tive homology.

Notice that we have Z[cusps] = C0/Z[R ∪ I], hence the quotientcomplex

0→ C2 ∂→ C1 ∂̃→ Z[cusps]→ 0,with ∂̃ the induced boundary operator, computes the relative homologyH1(XG, R ∪ I). The cycles are given by Z[P], as combinations of ele-ments g〈i, ρ〉, g ranging over representatives of P, and by the elements⊕ag〈g(i∞), g(i)〉 satisfying

∑

agg(i∞) = 0. In fact, these can be rep-resented as relative cycles in (XG, R ∪ I). Similarly, the subcomplex

0→ Z[P] ∂→ Z[R ∪ I]→ 0,with Z[P] generated by the elements g〈i, ρ〉, computes the homologyH1(XG − cusps). The homology

H1(XG − cusps, R ∪ I) ∼= Z[P]


is generated by the relative cycles g〈i, ρ〉, g ranging over representativesof P.

With the notation (2.4), we consider the groups H cuspsR∪I , HR∪Icusps,

Hcusps, and Hcusps.We have a long exact sequence of relative homology

(2.9) 0→ Hcusps → HR∪Icusps(β̃R,β̃I)→ H0(R)⊕H0(I)→ Z→ 0,

with Hcusps and HR∪Icusps as above, and with

H0(I) ∼= Z[PI ], PI = 〈σ〉\P = G\ĨH0(R) ∼= Z[PR], PR = 〈τ〉\P = G\R̃,

that is,

(2.10) 0→ Hcusps → Z[P]→ Z[PR]⊕ Z[PI ]→ Z→ 0.In the case of HR∪Icusps and H

cuspsR∪I the pairing (2.5) gives the identifi-

cation of Z[P] and Z|P|, obtained by identifying the elements of P withthe corresponding delta functions. Thus, we can rewrite the sequence(2.10) as

(2.11) 0→ Hcusps → Z|P| (βR,βI)→ Z|PI | ⊕ Z|PR| → Z→ 0,where HcuspsR∪I

∼= Z|P|.In order to understand more explicitly the map (βR, βI) we give the

following equivalent algebraic formulation of the modular complex (cf.[68] [77]).

The homology group HR∪Icusps = Z[P] is generated by the images in XGof the geodesic segments gγ0 := g〈i, ρ〉, with g ranging over a chosensetof representatives of the coset space P.

We can identify (cf. [77]) the dual basis δs of HcuspsR∪I = Z

|P| withthe images in XG of the paths gη0, where for a chosen point z0 with0 < Re(z0) < 1/2 and |z0| > 1 the path η0 is given by the geodesic arcsconnecting ∞ to z0, z0 to τz0, and τz0 to 0. These satisfy

[gγ0] • [gη0] = 1[gγ0] • [hη0] = 0,

for gG 6= hG, under the intersection pairing (2.5)Then, in the exact sequence (2.11), the identification of H cusps with

Ker(βR, βI) is obtained by the identification {g(0), g(i∞)}G 7→ gη0,so that the relations imposed on the generators δs by the vanishingunder βI correspond to the relations δs ⊕ δσs (or δs if s = σs) and thevanishing under βR gives another set of relations δs ⊕ δτs ⊕ δτ2s (or δsif s = τs).


2. The noncommutative boundary of modular curves

The main idea that bridges between the algebro–geometric theoryof modular curves and noncommutative geometry consists of replacingP1(Q) in the classical compactification, which gives rise to a finite setof cusps, with P1(R). This substitution cannot be done naively, sincethe quotient G\P1(R) is ill behaved topologically, as G does not actdiscretely on P1(R).

When we regard the quotient Γ\P1(R), or more generally Γ\(P1(R)×P), itself as a noncommutative space, we obtain a geometric object thatis rich enough to recover many important aspects of the classical theoryof modular curves. In particular, it makes sense to study in terms ofthe geometry of such spaces the limiting behavior for certain arithmeticinvariants defined on modular curves when τ → θ ∈ R \Q.

3. Modular interpretation: noncommutative elliptic curves

The boundary Γ\P1(R) of the modular curve Γ\H2, viewed itself asa noncommutative space, continues to have a modular interpretation,as observed originally by Connes–Douglas–Schwarz ([25]). In fact, wecan think of the quotients of S1 by the action of rotations by an irra-tional angle (that is, the noncommutative tori) as particular degener-ations of the classical elliptic curves, which are “invisible” to ordinaryalgebraic geometry. The quotient space Γ\P1(R) classifies these non-commutative tori up to Morita equivalence ([15], [85]) completes themoduli space Γ\H2 of the classical elliptic curves. Thus, from a concep-tual point of view it is reasonable to think of Γ\P1(R) as the boundaryof Γ\H2, when we allow points in this classical moduli space (that is,elliptic curves) to have non-classical degenerations to noncommutativetori.

Noncommutative tori are, in a sense, a prototype example of non-commutative spaces, in as one can see there displayed the full rangeof techniques of noncommutative geometry (cf. [15], [17]). As C∗–algebras, noncommutative tori are irrational rotation algebras. We re-call some basic properties of noncommutative tori, which justify theclaim that these algebras behave like a noncommutative version of el-liptic curves. We follow mostly [15] [20] and [85] for this material.

3.1. Irrational rotation and Kronecker foliation.

Definition 3.1. The irrational rotation algebra Aθ, for a givenθ ∈ R, is the universal C∗-algebra C∗(U, V ), generated by two unitaryoperators U and V , subject to the commutation relation

(2.12) UV = e2πiθV U.

3. MODULAR INTERPRETATION: NONCOMMUTATIVE ELLIPTIC CURVES25

The algebra Aθ can be realized as a subalgebra of bounded operatorson the Hilbert spaceH = L2(S1), with the circle S1 ∼= R/Z. For a givenθ ∈ R, we consider two operators, that act on a complete orthonormalbasis en of H as(2.13) Uen = en+1, V en = e

2πinθen.

It is easy to check that these operators satisfy the commutation re-lation (2.12), since, for any f ∈ H we have V Uf (t) = Uf(t − θ) =e2πi(t−θ)f(t− θ), while UV f (t) = e2πitf(t− θ).

The irrational rotation algebra be described in more geometricterms by the foliation on the usual commutative torus by lines withirrational slope. On T 2 = R2/Z2 one considers the foliation dx = θdy,for x, y ∈ R/Z. The space of leaves is described as X = R/(Z + θZ) 'S1/θZ. This quotient is ill behaved as a classical topological space,hence it cannot be well described by ordinary geometry.

A transversal to the foliation is given for instance by the choiceT = {y = 0}, T ∼= S1 ∼= R/Z. Then the non-commutative torus isobtained (cf. [15] [20]) as

(2.14) Aθ = {(fab) a, b ∈ T in the same leaf }where (fab) is a power series b =

∑

n∈Z bnVn and each bn is an element

of the algebra C(S1). The multiplication is given by

V hV −1 = h ◦R−1θ ,with

Rθx = x + θ mod 1.

The algebra C(S1) is generated by U(t) = e2πit, hence we recover thegenerating system (U, V ) with the relation

UV = e2πiθV U.

What we have obtained through this description is an identificationof the irrational rotation algebra of Definition 3.1 with the crossedproduct C∗-algebra

(2.15) Aθ = C(S1) oRθ Zrepresenting the quotient S1/θZ as a non-commutative space.

3.2. Degenerations of elliptic curves. An elliptic curve Eτ overC can be described as the quotient Eτ = C/(Z + τZ) of the complexplane by a 2-dimensional lattice Λ = Z+τZ, where we can take =(τ) >0. It is also possible to describe the elliptic curve Eq, for q ∈ C∗,q = exp(2πiτ), |q| < 1, in terms of its Jacobi uniformization, namely


��

��

|q| 1

Figure 4. The fundamental domain for the Jacobi uni-formization of the elliptic curve

as the quotient of C∗ by the action of the group generated by a singlehyperbolic element in PSL(2, C),

(2.16) Eq = C∗/qZ.

The fundamental domain for the action of qZ is an annulus

{z ∈ C : |q| < z ≤ 1}

of radii 1 and |q|, and the identification of the two boundary circles isobtained via the combination of scaling and rotation given by multi-plication by q.

Now let us consider a degeneration where q → exp(2πiθ) ∈ S1,with θ ∈ R \Q. We can say heuristically that in this degeneration theelliptic curve becomes a non-commutative torus

Eq =⇒ Aθ,

in the sense that, as we let q → exp(2πiθ), the annulus of the funda-mental domain shrinks to a circle S1 and we are left with a quotientof S1 by the infinite cyclic group generated by the irrational rotationexp(2πiθ). Since this quotient is ill behaved as a classical quotient,such degenerations do not admit a description within the context ofclassical geometry. However, when we replace the quotient by the cor-responding crossed product algebra C(S1) oθ Z we find the irrationalrotation algebra of definition 3.1. Thus, we can consider such algebrasas non-commutative (degenerate) elliptic curves.

3. MODULAR INTERPRETATION: NONCOMMUTATIVE ELLIPTIC CURVES27

More precisely, when one considers degenerations of elliptic curvesEτ = C

∗/qZ for q = e2πiτ , what one obtains in the limit is the suspen-sion of a noncommutative torus. In fact, as the parameter q degeneratesto a point on the unit circle, q → e2πiθ, the “nice” quotient Eτ = C∗/qZdegenerates to the “bad” quotient Eθ = C

∗/e2πiτZ, whose noncommu-tative algebra of coordinates is Morita equivalent Aθ ⊗ C0(R), withρ ∈ R the radial coordinate C∗ 3 z = eρe2πis.

Because of the Thom isomorphism [16], the K-theory of the non-commutative space Eθ = C0(R

2) o (Zθ + Z) satisfies

(2.17) K0(Eθ) = K1(Aθ) and K1(Eθ) = K0(Aθ),

which is again compatible with the identification of the Eθ (rather thanAθ) as degenerations of elliptic curves. In fact, for instance, the Hodgefiltration on the H1 of an elliptic curve and the equivalence between theelliptic curve and its Jacobian, have analogs for the noncommutativetorus Aθ in terms of the filtration on HC0 induced by the inclusion ofK0 (cf. [17] p. 132–139, [23] §XIII), while by the Thom isomorphism,these would again appear on the HC1 in the case of the “noncommu-tative elliptic curve” Eθ.

The point of view of degenerations is sometimes a useful guide-line. For instance, one can study the limiting behavior of arithmeticinvariants defined on the parameter space of elliptic curves (on modu-lar curves), in the limit when τ → θ ∈ R \Q. An instance of this typeof result is the theory of limiting modular symbols of [70].

3.3. Morita equivalent NC tori. To extend the modular in-terpretation of the quotient Γ\H2 as moduli of elliptic curves to thenoncommutative boundary Γ\P1(R), one needs to check that pointsin the same orbit of the action of the modular group PSL(2, Z) byfractional linear transformations on P1(R) define equivalent noncom-mutative tori, where equivalence here is to be understood in the Moritasense.

Connes showed in [15] (cf. also [85]) that the noncommutative toriAθ and A−1/θ are Morita equivalent. Geometrically in terms of theKronecker foliation and the description (2.14) of the corresponding al-gebras, the Morita equivalence Aθ ' A−1/θ corresponds to changingthe choice of the transversal from T = {y = 0} to T ′ = {x = 0}.

In fact, all Morita equivalences arise in this way, by changing thechoice of the transversal of the foliation, so that Aθ and Aθ′ are Moritaequivalent if and only if θ ∼ θ′, under the action of PSL(2, Z).


Connes constructed in [15] explicit bimodules realizing the Moritaequivalences between non-commutative tori Aθ and Aθ′ with

θ′ =aθ + b

cθ + d= gθ,

for

g =

(

a bc d

)

∈ Γ,

by taking Mθ,θ′ to be the Schwartz space S(R × Z/c), with the rightaction of Aθ

Uf (x, u) = f

(

x− cθ + dc

, u− 1)

V f (x, u) = exp(2πi(x− ud/c))f(x, u)and the left action of Aθ′

U ′f (x, u) = f

(

x− 1c, u− a

)

V ′f (x, u) = exp

(

2πi

(

x

cθ + d− u

c

))

f(x, u).

3.4. Other properties of NC elliptic curves. There are otherways in which the irrational rotation algebra behaves much like anelliptic curve, most notably the relation between the elliptic curve andits Jacobian (cf. [17] and [23]) and some aspects of the theory of thetafunctions, which we recall briefly.

The commutative torus T 2 = S1× S1 is connected, hence the alge-bra C(T 2) does not contain interesting projections. On the contrary,the noncommutative tori Aθ contain a large family of nontrivial pro-jections. Rieffel in [85] showed that, for a given θ irrational and forall α ∈ (Z ⊕ Zθ) ∩ [0, 1], there exists a projection Pα in Aθ, withTr(Pα) = α. A different construction of projections in Aθ, given byBoca [7], has arithmetic relevance, in as these projections correspondto the theta functions for noncommutative tori defined by Manin in[64].

A method of constructing projections in C∗-algebras is based onthe following two steps (cf. [85] and [63]):

(1) Suppose given a bimodule AMB. If an element ξ ∈ AMBadmits an invertible ∗-invariant square root 〈ξ, ξ〉1/2B , then theelement µ := ξ〈ξ, ξ〉−1/2B satisfies µ〈µ, µ〉B = µ.

(2) Let µ ∈ AMB be a non-trivial element such that µ〈µ, µ〉B = µ.Then the element P :=A 〈µ, µ〉 is a projection.

4. LIMITING MODULAR SYMBOLS 29

In Boca’s construction, one obtains elements ξ from Gaussian ele-ments in some Heisenberg modules, in such a way that the correspond-ing 〈ξ, ξ〉B is a quantum theta function in the sense of Manin [64]. Anintroduction to the relation between the Heisenberg groups and thetheory of theta functions is given in the third volume of Mumford’sTata lectures on theta, [79].

4. Limiting modular symbols

We consider the action of the group Γ = PGL(2, Z) on the up-per and lower half planes H± and the modular curves defined by thequotient XG = G\H±, for G a finite index subgroup of Γ. Then thenoncommutative compactification of the modular curves is obtained byextending the action of Γ on H± to the action on the full

P1(C) = H± ∪ P1(R),so that we have

(2.18) XG = G\P1(C) = Γ\(P1(C)× P).Due to the fact that Γ does not act discretely on P1(R), the quotient(2.18) makes sense as a noncommutative space

(2.19) C(P1(C)× P) o Γ.Here P1(R) ⊂ P1(C) is the limit set of the group Γ, namely the

set of accumulation points of orbits of elements of Γ on P1(C). Wewill see another instance of noncommutative geometry arising from theaction of a group of Möbius transformations of P1(C) on its limit set,in the context of the geometry at the archimedean primes of arithmeticvarieties.

4.1. Generalized Gauss shift and dynamics. We have de-scribed the boundary of modular curves by the crossed product C∗-algebra

(2.20) C(P1(R)× P) o Γ.We can also describe the quotient space Γ\(P1(R)×P) in the followingequivalent way. If Γ = PGL(2, Z), then Γ-orbits in P1(R) are the sameas equivalence classes of points of [0, 1] under the equivalence relation

x ∼T y ⇔ ∃n, m : T nx = T mywhere Tx = 1/x−[1/x] is the classical Gauss shift of the continued frac-tion expansion. Namely, the equivalence relation is that of having thesame tail of the continued fraction expansion (shift-tail equivalence).

A simple generalization of this classical result yields the following.


Lemma 4.1. Γ orbits in P1(R) × P are the same as equivalenceclasses of points in [0, 1]× P under the equivalence relation(2.21) (x, s) ∼T (y, t)⇔ ∃n, m : T n(x, s) = T m(y, t)where T : [0, 1]× P→ [0, 1]× P is the shift

(2.22) T (x, s) =

(

1

x−[

1

x

]

,

(

−[1/x] 11 0

)

· s)

generalizing the classical shift of the continued fraction expansion.

As a noncommutative space, the quotient by the equivalence re-lation (2.21) is described by the C∗-algebra of the groupoid of theequivalence relation

G([0, 1]× P, T ) = {((x, s), m− n, (y, t)) : T m(x, s) = T n(y, t)}with objects G0 = {((x, s), 0, (x, s))}.

In fact, for any T -invariant subset E ⊂ [0, 1]× P, we can considerthe equivalence relation (2.21). The corresponding groupoid C∗-algebraC∗(G(E, T )) encodes the dynamical properties of the map T on E.

Geometrically, the equivalence relation (2.21) on [0, 1]×P is relatedto the action of the geodesic flow on the horocycle foliation on themodular curves.

4.2. Arithmetic of modular curves and noncommutative

boundary. The result of Lemma 4.1 shows that the properties of thedynamical system T or (2.21) can be used to describe the geometry ofthe noncommutative boundary of modular curves. There are varioustypes of results that can be obtained by this method ([70] [71]), whichwe will discuss in the rest of this chapter.

(1) Using the properties of this dynamical system it is possibleto recover and enrich the theory of modular symbols on XG,by extending the notion of modular symbols from geodesicsconnecting cusps to images of geodesics in H2 connecting ir-rational points on the boundary P1(R). In fact, the irrationalpoints of P1(R) define limiting modular symbols. In the caseof quadratic irrationalities, these can be expressed in terms ofthe classical modular symbols and recover the generators ofthe homology of the classical compactification by cusps XG.In the remaining cases, the limiting modular symbols vanishesalmost everywhere.

(2) It is possible to reinterpret Dirichlet series related to modu-lar forms of weight 2 in terms of integrals on [0, 1] of certainintersection numbers obtained from homology classes defined


in terms of the dynamical system. In fact, even when thelimiting modular symbol vanishes, it is possible to associate anon-trivial cohomology class in XG to irrational points on theboundary, in such a way that an average of the correspondingintersection numbers give Mellin transforms of modular formsof weight 2 on XG.

(3) The Selberg zeta function of the modular curve can be ex-pressed as a Fredholm determinant of the Perron-Frobeniusoperator associated to the dynamical system on the “bound-ary”.

(4) Using the first formulation of the boundary as the noncom-mutative space (2.20) we can obtain a canonical identificationof the modular complex with a sequence of K-groups of theC∗-algebra. The resulting exact sequence for K-groups can beinterpreted, using the description (2.21) of the quotient space,in terms of the Baum–Connes assembly map and the Thomisomorphism.

All this shows that the noncommutative space C(P1(R) × P) o Γ,which we have so far considered as a boundary stratum of C(H2×P)oΓ, in fact contains a good part of the arithmetic information on theclassical modular curve itself. The fact that information on the “bulkspace” is stored in its boundary at infinity can be seen as an instance ofthe physical principle of holography (bulk/boundary correspondence)in string theory (cf. [69]). We will discuss the holography principlemore in details in relation to the geometry of the archimedean fibersof arithmetic varieties.

4.3. Limiting modular symbols. Let γβ be an infinite geodesicin the hyperbolic plane H with one end at i∞ and the other end atβ ∈ R r Q. Let x ∈ γβ be a fixed base point, τ be the geodesic arclength, and y(τ) be the point along γβ at a distance τ from x, towardsthe end β. Let {x, y(τ)}G denote the homology class in XG determinedby the image of the geodesic arc 〈x, y(τ)〉 in H.

Definition 4.1. The limiting modular symbol is defined as

(2.23) {{∗, β}}G := lim1

τ{x, y(τ)}G ∈ H1(XG, R),

whenever such limit exists.

The limit (2.23) is independent of the choice of the initial point xas well as of the choice of the geodesic in H ending at β, as discussedin [70] (cf. Figure 5). We use the notation {{∗, β}}G as introduced in


[70], where ∗ in the first argument indicates the independence on thechoice of x, and the double brackets indicate the fact that the homologyclass is computed as a limiting cycle.

4.3.1. Dynamics of continued fractions. As above, we consider on[0, 1]× P the dynamical system

T : [0, 1]× P→ [0, 1]× P

(2.24) T (β, t) =

(

1

β−[

1

β

]

,

(

−[1/β] 11 0

)

· t)

.

This generalizes the classical shift map of the continued fraction

T : [0, 1]→ [0, 1] T (x) = 1x−[

1

x

]

.

Recall the following basic notation regarding continued fraction ex-pansion. Let k1, . . . , kn be independent variables and, for n ≥ 1, let

[k1, . . . , kn] :=1

k1 +1

k2+...1

kn

=Pn(k1, . . . , kn)

Qn(k1, . . . , kn).

The Pn, Qn are polynomials with integral coefficients, which can becalculated inductively from the relations

Qn+1(k1, . . . , kn, kn+1) = kn+1Qn(k1, . . . , kn) + Qn−1(k1, . . . , kn−1),

Pn(k1, . . . , kn) = Qn−1(k2, . . . , kn),

with Q−1 = 0, Q0 = 1. Thus, we obtain

[k1, . . . , kn−1, kn + xn]

=Pn−1(k1, . . . , kn−1) xn + Pn(k1, . . . , kn)

Qn−1(k1, . . . , kn−1) xn + Qn(k1, . . . , kn)=

(

Pn−1 PnQn−1 Qn

)

(xn),

with the standard matrix notation for fractional linear transformations,

z 7→ az + bcz + d

=

(

a bc d

)

(z).

If α ∈ (0, 1) is an irrational number, there is a unique sequenceof integers kn(α) ≥ 1 such that α is the limit of [ k1(α), . . . , kn(α) ] asn→∞. Moreover, there is a unique sequence xn(α) ∈ (0, 1) such that

α = [ k1(α), . . . , kn−1(α), kn(α) + xn(α) ]

for each n ≥ 1. We obtain

α =

(

0 11 k1(α)

)

. . .

(

0 11 kn(α)

)

(xn(α)).


We set

pn(α) := Pn(k1(α), . . . , kn(α)), qn(α) := Qn(k1(α), . . . , kn(α))

so that pn(α)/qn(α) is the sequence of convergents to α. We also set

gn(α) :=

(

pn−1(α) pn(α)qn−1(α) qn(α)

)

∈ GL(2, Z).

Written in terms of the continued fraction expansion, the shift T isgiven by

T : [k0, k1, k2, . . .] 7→ [k1, k2, k3, . . .].

The properties of the shift (2.24) can be used to extend the no-tion of modular symbols to geodesics with irrational ends ([70]). Suchgeodesics correspond to infinite geodesics on the modular curve XG,which exhibit a variety of interesting possible behaviors, from closedgeodesics to geodesics that approximate some limiting cycle, to geodesicsthat wind around different homology class exhibiting a typically chaoticbehavior.

4.3.2. Lyapunov spectrum. A measure of how chaotic a dynamicalsystem is, or better of how fast nearby orbits tend to diverge, is givenby the Lyapunov exponent.

Definition 4.2. the Lyapunov exponent of T : [0, 1] → [0, 1] isdefined as

(2.25) λ(β) := limn→∞

1

nlog |(T n)′(β)| = lim

n→∞

1

nlog

n−1∏

k=0

|T ′(T kβ)|.

The function λ(β) is T–invariant. Moreover, in the case of the clas-sical continued fraction shift Tβ = 1/β − [1/β] on [0, 1], the Lyapunovexponent is given by

(2.26) λ(β) = 2 limn→∞

1

nlog qn(β),

with qn(β) the successive denominators of the continued fraction ex-pansion.

In particular, the Khintchine–Lévy theorem shows that, for almostall β’s (with respect to the Lebesgue measure on [0, 1]) the limit (2.26)is equal to

(2.27) λ(β) = π2/(6 log 2) =: λ0.


There is, however, an exceptional set in [0, 1] of Hausdorff dimensiondimH = 1 but with Lebesgue measure zero where limit defining theLyapunov exponent does not exist.

As we will see later, in “good cases” the value λ(β) can be computedfrom the spectrum of Perron–Frobenius operator of the shift T .

The Lyapunov spectrum is introduced (cf. [83]) by decomposing theunit interval in level sets of the Lyapunov exponent λ(β) of (2.25). LetLc = {β ∈ [0, 1] |λ(β) = c ∈ R}. These sets provide a T–invariantdecomposition of the unit interval,

[0, 1] =⋃

c∈R

Lc ∪ {β ∈ [0, 1] |λ(β) does not exist}.

These level sets are uncountable dense T–invariant subsets of [0, 1], ofvarying Hausdorff dimension [83]. The Lyapunov spectrum measureshow the Hausdorff dimension varies, as a function h(c) = dimH(Lc).

4.3.3. Limiting modular symbols and iterated shifts. We introducea function ϕ : P→ Hcusps of the form(2.28) ϕ(s) = {g(0), g(i∞)}G,where g ∈ PGL(2, Z) (or PSL(2, Z)) is a representative of the cosets ∈ P.

Then we can compute the limit (2.23) in the following way.

Theorem 4.2. Consider a fixed c ∈ R which corresponds to somelevel set Lc of the Lyapunov exponent (2.26). Then, for all β ∈ Lc, thelimiting modular symbol (2.23) is computed by the limit

(2.29) limn→∞

1

cn

n∑

k=1

ϕ ◦ T k(t0),

where T is the shift of (2.24) and t0 ∈ P.Without loss of generality, one can consider the geodesic γβ in H×P

with one end at (i∞, t0) and the other at (β, t0), for ϕ(t0) = {0, i∞}G.The argument given in §2.3 of [70] is based on the fact that one

can replace the homology class defined by the vertical geodesic withone obtained by connecting the successive rational approximations toβ in the continued fraction expansion (Figure 5). Namely, one canreplace the path 〈x0, yn〉 with the union of arcs

〈x0, y0〉 ∪ 〈y0, p0/q0〉 ∪n⋃

k=1

〈pk−1/qk−1, pk/qk〉 ∪ 〈pn/qn, yn〉


zn

zn+1

ζn

pn+1qn+1

pn-1

q n-1

pn qn

ζn+1

α β

Figure 5. Geodesics defining limiting modular symbols

representing the same homology class in H1(XG, Z).The result then follows by estimating the geodesic distance τ ∼

− log=y + O(1), as y(τ)→ β and1

2qnqn+1< =yn <

1

2qnqn−1,

where yn is the intersection of γβ and the geodesic with ends at pn−1(β)/qn−1(β)and pn(β)/qn(β).

The matrix g−1k (β), with

gk(β) =

(

pk−1(β) pk(β)qk−1(β) qk(β)

)

,

acts on points (β, t) ∈ [0, 1]× P as the k–th power of the shift map Tof (2.24). Thus, we obtain

ϕ(T kt0) = {g−1k (β) (0), g−1k (β) (i∞)}G ={

pk−1(β)

qk−1(β),pk(β)

qk(β)

}

G

.

4.4. Ruelle and Perron–Frobenius operators. A general prin-ciple in the theory of dynamical systems is that one can often studythe dynamical properties of a map T (e.g. ergodicity) via the spectraltheory of an associated operator. This allows one to employ techniquesof functional analysis and derive conclusions on dynamics.

In our case, to the shift map T of (2.24), we associate the operator

(2.30) (Lσf)(x, t) =∞∑

k=1

1

(x + k)2σf

(

1

x + k,

(

0 11 k

)

· t)

depending on a complex parameter σ.


More generally, the Ruelle transfer operator of a map T is definedas

(2.31) (Lσf)(x, t) =∑

(y,s)∈T−1(x,t)

exp(h(y, s)) f(y, s),

where we take h(x, t) = −2σ log |T ′(x, t)|. Clearly this operator is wellsuited for capturing the dynamical properties of the map T as it isdefined as a weighted sum over preimages. On the other hand, thereis another operator that can be associated to a dynamical system andwhich typically has better spectral properties, but is less clearly relatedto the dynamics. The best circumstances are when these two agree (fora particular value of the parameter). The other operator is the Perron–Frobenius operator P. This is defined by the relation

(2.32)

∫

[0,1]×P

f (g ◦ T ) dµLeb =∫

[0,1]×P

(P f) g dµLeb.

In the case of the shift T of (2.24) we have in fact that

P = Lσ|σ=1.

4.4.1. Spectral theory of L1. In the case of the modular group G =Γ, the spectral theory of the Perron Frobenius operator of the Gaussshift was studied by D.Mayer [75]. More recently, Chang and Mayer[13] extended the results to the case of congruence subgroups. A similarapproach is used in [70] to study the properties of the shift (2.24).

The Perron–Frobenius operator

(L1f)(x, s) =∞∑

k=1

1

(x + k)2f

(

1

x + k,

(

0 11 k

)

· s)

for the shift (2.24) has the following properties.

Theorem 4.3. On a Banach space of holomorphic functions onD×P continuous to boundary, with D = {z ∈ C | |z−1| < 3/2}, underthe condition (irreducibility)

(2.33) P = ∪∞n=0{(

0 11 k1

)

. . .

(

0 11 kn

)

(t0) | k1, . . . , kn ≥ 1}

,

the Perron–Frobenius operator L1 has the following properties:

• L1 is a nuclear operator, of trace class.• L1 has top eigenvalue λ = 1. This eigenvalue is simple. The

corresponding eigenfunction is (up to normalization)

1

(1 + x).


• The rest of the spectrum of L1 is contained in a ball of radiusr < 1.• There is a complete set of eigenfunctions.

The irreducibility condition (2.33) is satisfied by congruence sub-groups.

For other T -invariant subsets E ⊂ [0, 1]× P, one can also consideroperators LE,σ and PE. When the set has the property that PE =LE,δE , for δE = dimH E the Hausdorff dimension, one can use thespectral theory of the operator PE to study the dynamical propertiesof T .

The Lyapunov exponent can be read off the spectrum of the familyof operators LE,σ.

Lemma 4.4. Let λσ denote the top eigenvalue of LE,σ. Then

λ(β) =d

dσλσ|σ=dimH(E) µH a.e. in E

4.4.2. The Gauss problem. Let

(2.34) mn(x) := measure of {α ∈ (0, 1) | xn(α) ≤ x }with α = [ a1(α), . . . , an−1(α), an(α) + xn(α) ].

The asymptotic behavior of the measures mn is a famous problemon the distribution of continued fractions formulated by Gauss, whoconjectured that

(2.35) m(x) = limn→∞

mn(x)?=

1

log 2log(1 + x).

The convergence of (2.34) to (2.35) was only proved by R. Kuzmin in1928. Other proofs were then given by P. Lévy (1929), K. Babenko(1978) and D. Mayer (1991). The arguments used by Babenko andMayer use the spectral theory on the Perron–Frobenius operator. Ofthese different arguments only the latter extends nicely to the case ofthe generalized Gauss shift (2.24).

The Gauss problem can be formulated in terms of a recursive rela-tion

(2.36) m′n+1(x) = (L1 m′n)(x) :=

∞∑

k=1

1

(x + k)2m′n

(

1

x + k

)

.

The right hand side of (2.36) is the image of m′n under the Gauss–Kuzmin operator. This is nothing but the Perron–Frobenius operatorfor the shift T in the case of the group Γ = PGL(2, Z).

As a consequence of Theorem 4.3, one obtains the following result.


Theorem 4.5. Let

mn(x, t) := µLeb{(y, s) : xn(y) ≤ x, gn(y)−1(s) = t}.Then we have

• m′n(x, t) = Ln1 (1).• The limit m(x, t) = limn→∞ mn(x, t) exists and equals

m(x, t) =1

|P| log 2 log(1 + x).

This shows that there exists a unique T -invariant measure on [0, 1]×P. This is uniform in the discrete set P (the counting measure) and itis the Gauss measure of the shift of the continued fraction expansionon [0, 1].

4.5. Two theorems on limiting modular symbols. The resulton the T -invariant measure allows us to study the general behavior oflimiting modular symbols.

A special role is played by limiting modular symbols {{∗, β}} whereβ is a quadratic irrationality in R r Q.

Theorem 4.6. Let g ∈ G be hyperbolic, with eigenvalue Λg corre-sponding to the attracting fixed point α+g . Let Λ(g) := | log Λg|, and let` be the period of the continued fraction expansion of β = α+g . Then

{{∗, β}}G ={0, g(0)}G

Λ(g)

=1

λ(β)`

∑̀

k=1

{g−1k (β) · 0, g−1k (β) · i∞}G.

This shows that, in this case, the limiting modular symbols arelinear combinations of classical modular symbols, with coefficients inthe field generated over Q by the Lyapunov exponents λ(β) of thequadratic irrationalities.

In terms of geodesics on the modular curve, this is the case wherethe geodesic has a limiting cycle given by the closed geodesic {0, g(0)}G(Figure 6).

There is then the “generic case”, where, contrary to the previousexample, the geodesics wind around many different cycles in such away that the resulting homology class averages out to zero over longdistances (Figure 7).


Figure 6. Limiting modular symbols: limiting cycle

Figure 7. Limiting modular symbols: chaotic tanglingand untangling

Theorem 4.7. For a T -invariant E ⊂ [0, 1] × P, under the irre-ducibility condition for E,

Rτ (β, s) :=1

τ{x, y(τ)}G

converges weakly to zero. Namely, for all f ∈ L1(E, dµH),

limτ→∞

∫

E

Rτ (β, s) f(β, s) dµH(β, s) = 0.

This weak convergence can be improved to strong convergence µH(E)-almost everywhere. Thus, the limiting modular symbol satisfies

{{∗, β}}G ≡ 0 a.e. on E.This results depends upon the properties of the operator L1 and the

result on the T -invariant measure. In fact, to get the result on the weakconvergence ([70]) one notices that the limit (2.29) computing limitingmodular symbols can be evaluated in terms of a limit of iterates of thePerron–Frobenius operator, by

limn

1

λ0 n

n∑

k=1

∫

[0,1]×P

f(ϕ ◦ T k) = limn

1

λ0n

n∑

k=1

∫

[0,1]×P

(Lk1f)ϕ,

where λ(β) = λ0 a.e. in [0, 1].By the convergence of Lk11 to the density h of the T -invariant mea-

sure and of

Lk1f →(∫

fdµ

)

h,


this yields∫

[0,1]×P

{{∗, β}} f(β, t) dµ(β, t) =(∫

[0,1]×P

f dµ

) (∫

[0,1]×P

ϕ h dµ

)

=

(∫

f dµ

)

1

2#P

∑

s∈P

ϕ(s).

It is then easy to check that the sum∑

s∈P ϕ(s) = 0 since each term inthe sum changes sign under the action of the inversion σ ∈ PGL(2, Z)with σ2 = 1, but the sum is globally invariant under σ.

The argument extends to the case of other T -invariant subsets of[0, 1]×P for which the corresponding Perron–Frobenius operator LE,δEhas analogous properties (cf. [71]). The weak convergence provedby this type of argument can be improved to strong convergence byapplying the strong law of large numbers to the “random variables”ϕk = ϕ ◦ T k (cf. [71] for details). The result effectively plays the roleof an ergodic theorem for the shift T on E.

5. Hecke eigenforms

A very important question is what happens to modular forms at thenoncommutative boundary of the modular curves. There is a varietyof phenomena in the theory of modular forms that hint to the fact thata suitable class of “modular forms” survives on the noncommutativeboundary. Zagier introduced the term “quantum modular forms” todenote this important and yet not sufficiently understood class of exam-ples. Some aspects of modular forms “pushed” to the noncommutativeboundary were analyzed in [70], in the form of certain averages involv-ing modular symbols and Dirichlet series related to modular forms ofweight 2. We recall here the main results in this case.

We shall now consider the case of congruence subgroups G = Γ0(p),for p a prime.

Let ω be a holomorphic differential (also called a differential of thefirst kind) on the modular curve XΓ0(p) = Γ0(p)\H. Let Φ = ϕ∗(ω)/dzbe the pullback under the projection ϕ : H→ XΓ0(p).

Let Φ be an eigenfunction for all the Hecke operators

Tn =∑

d|n

d−1∑

b=0

(

n/d b0 d

)

with (p, n) = 1,

TnΦ = cnΦ.

5. HECKE EIGENFORMS 41

Then the L-function of ω is given by

Lω(s) = −(2π)s

(2πi)Γ(s)

∫ ∞

0

Φ(iy) ys−1 dy

Lω(s) =

∞∑

n=1

cn n−s.

There are many very interesting arithmetic properties of the inte-grals of such Hecke eigenforms on modular symbols (Manin [68]).

In particular, one has ([68]) the following relation between Lω andmodular symbols:

(2.37) (σ(n)− cn) Lω(1) =∑

d|n, b mod d

∫

{0,b/d}

ω,

with σ(n) =∑

d|n d, from which one obtains

(2.38)

∑

q:(q,p)=1 q−(2+t)

∑

q′≤q,(q,q′)=1

∫

{0,q′/q}ω

=[

ζ(p)(1+t)

ζ(p)(2+t)− L

(p)ω (2+t)

ζ(p)(2+t)2

]

∫ i∞

0φ∗(ω)

Here L(p)ω is the Mellin transform of Φ with omitted Euler p–factor,

and ζ(s) the Riemann zeta, with corresponding ζ (p), that is, L(p)ω (s) =

∑

n:(n,p)=1 cn n−s and ζ (p)(s) =

∑

n:(n,p)=1 n−s.

It is therefore interesting to study the properties of integrals onlimiting modular symbols (cf. [70]) and extensions of (2.37) (2.38).

To this purpose, in [70] a suitable class of functions on the boundaryP1(R)× P (or just [0, 1]× P) was introduced.

These are functions of the form

(2.39) `(f, β) =

∞∑

k=1

f(qk(β), qk−1(β)).

Here f is a complex valued function defined on pairs of coprime integers(q, q′) with q ≥ q′ ≥ 1 and with f(q, q′) = O(q−�) for some � > 0,and qk(β) are the successive denominators of the continued fractionexpansion of β ∈ [0, 1].

The reason for this choice of functions is given by the followingclassical result of Lévy (1929).

Proposition 5.1. For a function f as above, we have

(2.40)

∫ 1

0

`(f, α)dα =∑

′ f(q, q′)

q(q + q′).

Sums and integrals here converge absolutely and uniformly.


We can interpret the summing over pairs of successive denominatorsas a property that replaces modularity, when “pushed to the bound-ary”.

Through this class of functions it is possible recast certain averagesrelated to modular symbols on XG, completely in terms of functiontheory on the “boundary” space Γ\(P1(R)×P). A typical result of thissort is the following ([70]).

Upon choosing

f(q, q′) =q + q′

q1+t

∫

{0,q′/q}G

ω,

one obtains∫ 1

0

`(f, β) dµLeb(β) =

[

ζ(1 + t)

ζ(2 + t)− L

(p)ω (2 + t)

ζ(p)(2 + t)2

]

∫ i∞

0

φ∗(ω),

which expresses the Mellin transform of a Hecke eigenform in terms ofa boundary average.

Analogous results exist for different choices of f(q, q ′), for Eisensteinseries twisted by modular symbols, double logarithms at roots of unity,etc.

In fact, this result also shows that, even when the limiting modularsymbol vanishes, one can still associate nontrivial homology classes tothe geodesics with irrational ends.

In fact, for the case of G = Γ0(p) let us consider

(2.41) C(f, β) :=∞∑

n=1

qn+1(β) + qn(β)

qn+1(β)1+t

{

0,qn(β)

qn+1(β)

}

Γ0(p)

.

This defines, for 0 and for almost all β, a homology class inH1(XG, cusps, R) such that the integral average

`(f, β) =

∫

C(f,β)

ω

on [0, 1] recovers Mellin transforms of cusp forms by∫ 1

0

∫

C(f,β)

ω dµLeb(β) =

[

ζ(1 + t)

ζ(2 + t)− L

(p)ω (2 + t)

ζ(p)(2 + t)2

]

∫ i∞

0

φ∗(ω).

An estimate

qn+1(β) + qn(β)

qn+1(β)1+t

{

0,qn(β)

qn+1(β)

}

G

∼ e−(5+2t)nλ(β)n∑

k=1

ϕ ◦ T k(s)

6. SELBERG ZETA FUNCTION 43

shows that the series involved are absolutely convergent.

6. Selberg zeta function

Infinite geodesics on XG are the image image of infinite geodesics onH2×P with ends on P1(R)×P. Thus, they can be coded by (ω−, ω+, s),with (ω±, s) = endpoints in P1(R) × {s}, s ∈ P, ω− ∈ (−∞,−1],ω+ ∈ [0, 1].

If ω± are not cusp points, then we have infinite continued fractionexpansions of these endpoints,

ω+ = [k0, k1, . . . kr, kr+1, . . .]ω− = [k−1; k−2, . . . , k−n, k−n−1, . . .]

,

hence the corresponding geodesics are coded by (ω, s), s ∈ P, where ωis a doubly infinite sequence

ω = . . . k−(n+1)k−n . . . k−1k0k1 . . . kn . . .

The equivalence relation of passing to the quotient by the groupaction is implemented by the invertible (double sided) shift:

T (ω+, ω−, s) =

(

1

ω+−[

1

ω+

]

,1

ω− − [1/ω+] ,(

−[1/ω+] 11 0

)

· s)

.

In particular, the closed geodesics in XG correspond to the casewhere the endpoints ω± are the attractive and repelling fixed points ofa hyperbolic element in the group. This corresponds to the case where(ω, s) is a periodic point for the shift T .

The Selberg zeta function is a suitable “generating function” forthe closed geodesics in XG. It is given by

ZG(s) =

∞∏

m=0

∏

γ

(

1− e−(s+m)`(γ))

,

where γ is a primitive closed geodesic of length `(γ).This can also be expressed in terms of the generalized Gauss shift

by the following ([13], [70])

Theorem 6.1. The Selberg zeta function for G ⊂ PGL(2, Z) offinite index is computed by the Fredholm determinant of the Ruelletransfer operator (2.31),

ZG(s) = det(1− Ls).An analogous result holds for finite index subgroups of SL(2, Z),

where one gets det(1− L2s). There are also generalizations to other T -invariant sets E, where det(1−LE,s) corresponds to a suitable “Selbergzeta” that only counts certain classes of closed geodesics.


Figure 8. The modular complex and the tree ofPSL(2, Z)

7. The modular complex and K-theory of C∗-algebras

When we interpret the “boundary” P1(R)× P with the action of Γas the noncommutative space C∗-algebra C(P1(R)×P)oΓ, it is possibleto reinterpret some of the arithmetic properties of modular curves interms of these operator algebras. For instance, the modular symbolsdetermine certain elements in the K-theory of this C∗-algebra, and themodular complex and the exact sequence of relative homology (2.11)can be identified canonically with the Pimsner six terms exact sequencefor K–theory of this C∗-algebra.

For Γ = PSL(2, Z) and G ⊂ Γ a finite index subgroup, we use thenotation X := P1(R)× P, and Γσ := 〈σ〉 = Z/2 and Γτ := 〈τ〉 = Z/3.

The group Γ acts on the tree with edges T 1 ' Γ and verticesT 0 ' Γ/〈σ〉 ∪ Γ/〈τ〉. This tree is embedded in H2 with vertices atthe elliptic points and geodesic edges, as the dual graph of the trian-gulation (Figure 3) of the modular complex (Figure 8).

For a group acting on a tree there is a Pimsner six terms exactsequence, computing the K-theory of the crossed product C∗-algebra(for A = C(X))

K0(A) α // K0(Ao Γσ)⊕K0(Ao Γτ ) α̃ // K0(Ao Γ)

��K1(Ao Γ)

OO

K1(Ao Γσ)⊕K1(Ao Γτ )β̃

oo K1(A)β

oo

A direct inspection of the maps in this exact sequence shows thatit contains a subsequence, which is canonically isomorphic to the alge-braic presentation of the modular complex, compatibly with the cover-ing maps between modular curves for different congruence subgroups.

8. INTERMEZZO: CHAOTIC COSMOLOGY 45

Theorem 7.1. The algebraic presentation of the modular complex

0→ Hcusps → Z[P]→ Z[PR]⊕ Z[PI ]→ Z→ 0is canonically isomorphic to the sequence

0→ Ker(β)→ Z[P]→ Z[PI ]⊕ Z[PR]→ Im(β̃)→ 0.A more geometric explanation for this exact sequence is the follow-

ing. The Lie group PSL(2, R) can be identified with the circle bundle(unit sphere bundle) over the hyperbolic plane H = PSL(2, R)/PSO(2).Thus, the unit tangent bundle to the modular curve XG is given by thequotient T1XG = G\PSL(2, R).

On the other hand, we can identify P1(R) = B\PSL(2, R), thequotient by the upper triangular matrices B, so that we have Moritaequivalent algebras

C(P1(R)) o G ' C(T1XG) o B.The right hand side is the semidirect product of two R-actions. Thus,by the Thom isomorphism [16] we have

K∗(C(T1XG) o B) ∼= K∗(C(T1XG))where K∗(T1XG) is related to K

∗(XG) by the Gysin exact sequence.A construction of crucial importance in noncommutative geometry

is the classifying space for proper actions BG of Baum–Connes [6], thehomotopy quotient X ×G EG and the µ-map

µ : K∗(X ×G EG)→ K∗(C(X) o G)relating the topologically defined K-theory to the analytic K-theory ofC∗-algebras.

In a sense, for noncommutative spaces that are obtained as quo-tients by “bad” equivalence relations, the homotopy quotient is a “com-mutative shadow” from which much crucial information on the topol-ogy can be read (cf. [6], [18]).

In our case we have EG = H2 and the µ-map is an isomorphism(the Baum–Connes conjecture holds). By retracting EG = H2 to thetree of PSL(2, Z) and applying the Mayer–Vietoris sequence to verticesand edges one obtains the Pimsner six terms exact sequence.

8. Intermezzo: Chaotic Cosmology

We digress momentarily on a topic from General Relativity, whichturns out to be closely related to the “noncommutative compactifica-tion” of the modular curve XG with G the congruence subgroup Γ0(2)([70]).


An important problem in cosmology is understanding how anisotropyin the early universe affects the long time evolution of space-time. Thisproblem is relevant to the study of the beginning of galaxy formationand in relating the anisotropy of the background radiation to the ap-pearance of the universe today.

We follow [5] for a brief summary of anisotropic and chaotic cosmol-ogy. The simplest significant cosmological model that presents stronganisotropic properties is given by the Kasner metric

(2.42) ds2 = −dt2 + t2p1dx2 + t2p2dy2 + t2p3dz2,where the exponents pi are constants satisfying

∑

pi = 1 =∑

i p2i .

Notice that, for pi = d log gii/d log g, the first constraint∑

i pi = 1 isjust the condition that log gij = 2αδij + βij for a traceless β, whilethe second constraint

∑

i p2i = 1 amounts to the condition that, in the

Einstein equations written in terms of α and βij,

(

dα

dt

)2

=8π

3

(

T 00 +1

16π

(

dβijdt

)2)

e−3αd

dt

(

e3αdβijdt

)

= 8π

(

Tij −1

3δijTkk

)

,

the term T 00 is negligible with respect to the term (dβij/dt)2/16π,

which is the “effective energy density” of the anisotropic motion ofempty space, contributing together with a matter term to the Hubbleconstant.

Around 1970, Belinsky, Khalatnikov, and Lifshitz introduced a cos-mological model (mixmaster universe) where they allowed the expo-nents pi of the Kasner metric to depend on a parameter u,

(2.43)

p1 =−u

1+u+u2

p2 =1+u

1+u+u2

p3 =u(1+u)1+u+u2

Since for fixed u the model is given by a Kasner space-time, the behav-ior of this universe can be approximated for certain large intervals oftime by a Kasner metric. In fact, the evolution is divided into Kasnereras and each era into cycles. During each era the mixmaster universegoes through a volume compression. Instead of resulting in a collapse,as with the Kasner metric, high negative curvature develops resultingin a bounce (transition to a new era) which starts again a behaviorapproximated by a Kasner metric, but with a different value of theparameter u. Within each era, most of the volume compression is due

8. INTERMEZZO: CHAOTIC COSMOLOGY 47

to the scale factors along one of the space axes, while the other scalefactors alternate between phases of contraction and expansion. Thesealternating phases separate cycles within each era.

More precisely, we are considering a metric

(2.44) ds2 = −dt2 + a(t)dx2 + b(t)dy2 + c(t)dz2,generalizing the Kasner metric (2.42). We still require that (2.44)admits SO(3) symmetry on the space-like hypersurfaces, and that itpresents a singularity at t→ 0. In terms of logarithmic time dΩ = − dt

abc,

the mixmaster universe model of Belinsky, Khalatnikov, and Lifshitzadmits a discretization with the following properties:

1. The time evolution is divided in Kasner eras [Ωn, Ωn+1], for n ∈ Z.At the beginning of each era we have a corresponding discrete value ofthe parameter un > 1 in (2.43).2. Each era, where the parameter u decreases with growing Ω, canbe subdivided in cycles corresponding to the discrete steps un, un − 1,un − 2, etc. A change u → u − 1 corresponds, after acting with thepermutation (12)(3) on the space coordinates, to changing u to −u,hence replacing contraction with expansion and conversely. Withineach cycle the space-time metric is approximated by the Kasner metric(2.42) with the exponents pi in (2.43) with a fixed value of u = un−k >1.3. An era ends when, after a number of cycles, the parameter unfalls in the range 0 < un < 1. Then the bouncing is given by thetransition u→ 1/u which starts a new series of cycles with new Kasnerparameters and a permutation (1)(23) of the space axis, in order tohave again p1 < p2 < p3 as in (2.43).

Thus, the transitio

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Lectures on Arithmetic Noncommutative Geometrymarcolli/BookVanderbilt.pdf · 2004. 9. 28. ·...

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